The weight enumerator of a binary doubly even self-dual code is an isobaric polynomial in the two generators of the ring of invariants of a certain group of order 192. The aim of this note is to study the ring of coefficients of that polynomial, both for standard and joint weight enumerators.

The Manin ring is a family of quadratic algebras describing pointed stable curves of genus zero whose homology gives the solution of the Commutativity Equations. This solution was first observed by the physicist Losev. We show the Manin ring is the Stanley–Reisner ring of the standard triangulation of the n-cube modulo a system of parameters. Thus, the Hilbert series of the Manin ring is given ...

Let A = K[x1, . . . , xn] denote the polynomial ring in n variables over a field K. We will classify all the Gotzmann ideals of A with at most n generators. In addition, we will study Hilbert functions H for which all homogeneous ideals of A with the Hilbert function H have the same graded Betti numbers. These Hilbert functions will be called inflexible Hilbert functions. We introduce the notio...

The structure of half-BPS representations psu$(2,2|4)$ leads to the definition a super-polynomial ring $\mathcal{R}(8|8)$ which admits realisation in terms differential operators on super-ring. character fundamental field representation encodes resolution an exact sequence modules $\mathcal{R}(8|8)$. is realized by quotienting super-ring quadratic ideal, equivalently setting zero certain polyno...

Given p polynomials with coefficients in a commutative unitary integral ring C containing Q, we define the notion of a generic Bernstein-Sato polynomial on an irreducible affine scheme V ⊂ Spec(C). We prove the existence of such a non zero rational polynomial which covers and generalizes previous existing results by H. Biosca. When C is the ring of an algebraic or analytic space, we deduce a st...

We construct reduced Groebner bases for a certain class of ideals in commutative polynomial rings. A subclass of these ideals corresponds to the generalized Reed-Muller codes when considered in the quotient ring of the polynomial ring. AMS Subject Classification: 13P10, 94B30

Let R = k[x1, . . . , xn] be a polynomial ring over a field k. We present a characterization of multigraded R-modules in terms of the minors of their presentation matrix. We describe explicitly the second syzygies of any multigraded R-module.

Consider the system of Diophantine equations x2 − ay2 = b, P (x, y) = z2, where P is a given integer polynomial. Historically, such systems have been analyzed by using Baker’s method to produce an upper bound on the integer solutions. We present a general elementary approach, based on an idea of Cohn and the theory of the Pell equation, that solves many such systems. We apply the approach to th...

Codes over finite quotient of polynomial rings

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x, y], which establishes a conjecture of Haiman and Sturmfels. © 2009 Gregory G. Smith. Published by Elsevier Inc. All rights reserved.